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Kronecker :: kroneckerNormalForm(Matrix,Matrix)

kroneckerNormalForm(Matrix,Matrix) -- normal form of a pair of matrices of scalars

Synopsis

Description

This function gives the normal form of a pair of matrices (A,B) over a field of the same dimensions up to multiplication on either side by an invertible matrix. The return values are such that P*A*Q=A' and P*B*Q=B'.
i1 : R = QQ

o1 = QQ

o1 : Ring
i2 : A = random(R^2, R^5)

o2 = | 1/3 3   3/5 7/9  1   |
     | 7/8 8/3 9   3/10 5/6 |

              2        5
o2 : Matrix QQ  <--- QQ
i3 : B = random(R^2, R^5)

o3 = | 4 1/5 8/3  1/5 7/6 |
     | 5 10  10/3 1   1   |

              2        5
o3 : Matrix QQ  <--- QQ
i4 : (A',B',P,Q) = kroneckerNormalForm(A,B)

o4 = (| 0 1 0 0 0 |, | 0 0 1 0 0 |, | 0         -1/14069925 |, | -13379176 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/562797 0           |  | 16381515  
                                                               | -5990655  
                                                               | -161369970
                                                               | 84419550  
     ------------------------------------------------------------------------
     1064520  -554596  44496    -125992 |)
     -91800   -2044185 81810    96195   |
     -1679400 399120   6885     -5835   |
     1193400  7814505  -1063530 -312540 |
     0        0        0        0       |

o4 : Sequence
i5 : P*A*Q - A' == 0

o5 = true
i6 : P*B*Q - B' == 0

o6 = true